Hom--Operators and Hom-Yang-Baxter Equations
In Hom-Lie set, we introduce the concept of Hom--operators and study its relation with classical Hom-Yang-Baxter equation, as well as left-symmetric Hom-algebras. We construct the corresponding relation between left-symmetric Hom-algebras and Hom-1-cocycles, which are both related to classical Hom-Yang-Baxter equation. Moreover, in Hom-algebra setting, we establish the equivalent relationship between AHYBE (associative Hom-Yang-Baxter equations) and -operators on Frobenius monoidal Hom-algebras.
Hom-type algebras appeared first in physical contexts, in connection with twisted, discretized, or deformed derivatives and corresponding generalizations, discretizations, and deformations of vector fields and differential calculus. Hom-Lie algebras have been first introduced within context of more general quasi-Lie algebras and quasi-Hom-Lie algebras. Discretization of vector fields via twisted derivations leads to quasi-Hom-Lie structures in which the Jacobi identity is twisted by linear maps in [1–4]. The first examples and constructions of quasi-Hom-Lie algebras and Hom-Lie algebras have been concerned with the -deformations of Witt and Virasoro algebras obtained when the derivations are replaced by -derivations . In Hom-Lie algebras, the Jacobi identity is replaced by the so-called Hom-Jacobi identity,where is an endomorphism of Lie algebras. Recently, various Hom-Lie structures have been studied further by many scholars in [5–12]. Among them are such structures as Hom-Lie bialgebras, quasi-Hom-Lie algebras, Hom-Lie superalgebras, Hom-Lie color algebras, Hom-Lie admissible Hom-algebras, and Hom-Nambu-Lie algebras. In , the construction of Hom-Lie bialgebras both from Hom-Lie algebras and Hom-Lie coalgebras was investigated, respectively. Quasi-triangular Hom-Lie bialgebras were considered further in .
The ideal of tailoring associativity-like conditions by endomorphisms was migrated to other algebraic structures. Hom-associative algebras, as an analogue and generalization of associative algebras for Hom-Lie algebras, have been introduced in . Then the concepts of Hom-algebras, Hom-coalgebras, Hom-Hopf algebras, Hom-alternative algebras, Hom-Jordan algebras, Hom-Poisson algebras, Hom-Leibniz algebras, infinitesimal Hom-bialgebras, Hom-power associative algebras, and quasi-triangular Hom-bialgebras were introduced and further developed in [8, 15–22]. The integral and smash products of monoidal Hom-Hopf algebras were introduced in . Further, some actions and coactions on these Hom-algebras structures such as Hom-modules, Hom-comodules, Hom-Hopf modules, and Yetter-Drinfel’d Hom-modules were considered in , and the fundamental structure theorem of Hom-Hopf modules was investigated in .
Hom-associative algebras and Hom-coassociative coalgebras were introduced in [8, 19]. Here the associativity of algebras and the coassociativity of coalgebras were twisted by endomorphisms. Hom-bialgebras are both Hom-associative algebras and Hom-coassociative coalgebras such that the comultiplication and counit are morphisms of algebras. These objects are slightly different from the ones (called monoidal Hom-structures) studied in this paper (see Section 5).
Yang-Baxter system plays a crucial role in many fields like integrable systems, quantum groups, quantum field, and so on and has become an important topic in both mathematics and mathematical physics since 1980s. Classical Yang-Baxter equation (CYBE) first arose in the study of inverse scattering theory. The different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified method in . It is closely related to left-symmetric algebras which play an important role in mathematics and mathematical physics. A direct relation between the -operators and CYBE was obtained in .
The Rota-Baxter operator has appeared in a wide range of areas both in mathematics and physics. As a generalization of Rota-Baxter operators, -operators also have a close relationship with the associative Yang-Baxter equation (AYBE). Bai introduced the extended -operator which generalizes the concept of -operators and studied the relation between the extended -operator and AYBE in .
The main purpose of this paper is to investigate these objects above, equations, and their relations above in Hom-setting. Explicitly, we study Hom--operators and their relation with classical Hom-Yang-Baxter equation, as well as left-symmetric Hom-algebras. On the other hand, the equivalent relationship between AHYBE and -operators (on Frobenius monoidal Hom-algebras) is obtained.
This paper is organized as follows. In Section 3, we introduce the concept of Hom--operators, which generalize -operators. And the relationship between Hom--operators and classical Hom-Yang-Baxter equation (CHYBE) is studied. In Section 4, we study the relations between left-symmetric Hom-algebras and CHYBE through Hom-1-cocycles. Left-symmetric Hom-algebras can be constructed from Hom-1-cocycles. Conversely, there is a Hom-1-cocycle associated with left-symmetric Hom-algebras. Further, the equivalent relations among Hom-1-cocycles, Hom--operators, and CHYBE are constructed. In Sections 5 and 6, we study similar operators related to monoidal Hom-algebras in Hom-category and establish the equivalent relation between AHYBE and -operators as well as the relationship between -operators (Rota-Baxter operators) and the AHYBE on Frobenius monoidal Hom-algebras.
Throughout, will be a fixed field and all vector spaces, tensor products, and homomorphisms are over . We use Sweedler’s notation for terminologies on coalgebras and comodules. For a coalgebra , we write its comultiplication , for any ; for a right -comodule , we denote its coaction by , for any , in which we often omit the summation symbols for convenience. denotes the transposition map.
For a Lie coalgebra , we write its Lie-cobracket , , for any ; for a left -comodule , we denote its location by for any . Let be the cyclic permutation ; we denote the symbol by the sum over . Namely, we denote the Hom-Jacobi identity by in place of , for any .
In what follows, we recall some concepts used in this paper.
Definition 1. A Hom-Lie algebra  is a triple consisting of vector space , bilinear map (called the “bracket”), and a linear endomorphism satisfyingfor any .
Let be a Lie algebra, and let be a Lie algebra endomorphism. Define a new bracket on by setting ; then a direct calculation shows that is a Hom-Lie algebra.
Definition 2. Let be a Hom-Lie algebra. An -Hom-Lie module consists of a vector space and a linear endomorphism together with a bilinear function , satisfying for all .
It is straightforward that Hom-Lie algebra is itself an -Hom-Lie module via its Lie-bracket .
Let be a Hom-Lie algebra. For any and integer number , we define the adjoint action by
In particular, for and , we have
Definition 3. A Hom-Lie coalgebra introduced in  is a triple with a vector space , linear map (called the “cobracket”), and a linear endomorphism , such that
Definition 4. Let be a Hom-Lie algebra and . One calls it the coboundary triangular Hom-Lie algebra, if it satisfies the classical Hom-Yang-Baxter equation (CHYBE for short) where where , , and .
3. Hom--Operator and Classical Hom-Yang-Baxter Equation
In this section, we introduce the concept of Hom--operators, which generalized that of -operators. And we also study the relationship between Hom--operators and classical Hom-Yang-Baxter equation.
Definition 5. Let be a Hom-Lie algebra and let be an -Hom-Lie module. A linear map is called a Hom--operator associated with if it satisfies the following two conditions:
Proposition 6. Let be a Hom-Lie algebra and let be an -Hom-Lie module with , . Then there is a natural Hom-Lie algebra structure on (denoted by ) given by for any , .
Proof. For any , , by the bracket of and the antisymmetry of . Next, , since is multiplicative and satisfies equality (4). The Hom-Jacobi identity is also true: for any , ,
Lemma 7. Let be a Hom-Lie algebra; let be an -Hom-Lie module of -dimension. And let be the dual of with , for . Then is also a Hom-Lie module with the structure if and only if , where is a basis of and is the corresponding dual basis.
Proof. By the definition of , for any , , and , so if and only if .
For any , Then if and only if implied by the compatibility condition (3). Thus is also a Hom-Lie module with the structure given as above if and only if .
If we assume that , then there is a Hom-Lie algebra .
Let be a basis of , let be a basis of , and let be its dual basis; that is, . Set by with , where (). Since as spaces, we have
Lemma 8. With the above assumption, the skew-symmetric part is a solution of CHYBE in if and only if is a Hom--operator, where .
Proof. In fact, the Lie-bracket on given in (12) and equality (18) imply that By the definition of , we have where the last step is from the fact that pairs and are equivalent to each other in the sense of dual basis.
ThusSo is a solution of CHYBE in if and only if is a Hom--operator.
In particular, set the Hom-Lie module action on Hom-Lie algebra . In this case, we suppose that the Hom-Lie algebra is equipped with a nondegenerate symmetric Hom-associative bilinear form . That is, Hence can be identified with . Let be skew-symmetric; that is, . Then is a solution of CHYBE in if and only if it satisfies the equation Thus, in this case, the CHYBE is equivalent to (23).
4. Left-Symmetric Hom-Algebras and CHYBE
In this section, we study the relations between left-symmetric Hom-algebras and CHYBE through Hom-1-cocycles. Left-symmetric Hom-algebras can be constructed from Hom-1-cocycle and, conversely, there are Hom-1-cocycles associated with left-symmetric Hom-algebras. As the main result of this section, we construct the equivalent relation among Hom-1-cocycles, Hom--operators, and CHYBE.
Definition 9. A left-symmetric Hom-algebra is a vector space over a field equipped with a bilinear product and an endomorphism such that is multiplicative; that is, , and the associator is symmetric in , for any ; that is, , or equivalently
Similarly, we can define right-symmetric Hom-algebras. A Hom-associative algebra in  is obviously a left-symmetric Hom-algebra since the associator is zero.
A left-symmetric Hom-algebra can define a Hom-Lie algebra via the commutator for any . Indeed, the skew-symmetry is obvious, and the Hom-Jacobi identity is also true by We denote this above induced Hom-Lie algebra by .
Let be the left multiplication operator; that is, . Then which gives an -Hom-Lie module on itself by the Hom-Jacobi identity and the symmetry of the associator of left-symmetric Hom-algebras.
Definition 10. Let be a Hom-Lie algebra, and let be an -Hom-Lie module. A Hom-1-cocycle of associated with (denoted by ) is defined as a linear map such that and for any .
Proposition 11. Let be a Hom-Lie algebra, and let be an -Hom-Lie module. If is bijective, then there is a compatible left-symmetric Hom-algebra structure on if and only if there exists a bijective Hom-1-cocycle of .
Proof. Let be a bijective Hom-1-cocycle of associated with . Then, for any , defines a compatible left-symmetric Hom-algebra structure on . In fact, Conversely, for a left-symmetric Hom-algebra with bijective , given in (28) is a bijective Hom-1-cocycle of . And the Hom-1-cocycle condition is followed by (26) and the multiplicativity of .
Note that, for any Hom-Lie algebra and an -Hom-Lie module , a linear isomorphism (hence ) is a Hom--operator associated with if and only if is a bijective Hom-1-cocycle of to . Indeed, is a Hom--operator if and only if, for all , if and only if if and only if for all , which means that is a Hom-1-cocycle of .
From the above discussion together with Lemma 8, we obtain the main result of this section.
Theorem 12. Let be a Hom-Lie algebra with , and let be an -Hom-Lie module. Then is a bijective Hom-1-cocycle of associated with if and only if is a Hom--operator associated with if and only if is a solution of CHYBE in .
In particular, if a Hom-Lie algebra is also a left-symmetric Hom-algebra with the invertible endomorphism , then is a Hom--operator associated with the action defined in (28) since , for any . So we have that is a solution of CHYBE in , where is a basis of and is the corresponding dual basis.
Theorem 13. Let be a Hom-Lie algebra with , and let be an -Hom-Lie module. If is an invertible Hom--operator associated with , then is an isomorphism of both left-symmetric Hom-algebras and Hom-Lie algebras.
Proof. If Hom--operator associated with is invertible, then is a bijective Hom-1-cocycle of associated with . Hence, for any , defines a compatible left-symmetric Hom-algebra structure on by Proposition 11. Moreover, for any , let , ; then we have Since is invertible, there is a left-symmetric Hom-algebra structure on induced from the left-symmetric Hom-algebra structure on by Then we obtain ; thus is an isomorphism of left-symmetric Hom-algebras between them.
On the other hand, let be a linear map with . Then for any , the product satisfies Hence (36) defines a left-symmetric Hom-algebra if and only if, for any , . In particular, for any Hom--operator associated with , (36) defines a left-symmetric Hom-algebra on . Further we can define a Hom-Lie algebra structure on by (26). And ; hence is also a Hom-Lie algebraic homomorphism.
5. Monoidal Hom-Algebras and Associative Hom-Yang-Baxter Equations
In this section, we give the equivalent relation between AHYBE and -operators over monoidal Hom-algebras.
Let be the monoidal category of -modules. There is a new monoidal category . The objects of are couples , where and . The morphisms of are morphisms in such that . For any objects , the monoidal structure is given by
Briefly speaking, all Hom-structures are objects in the monoidal category introduced in , where the associator is given by the formula for any objects , and the unitors and are The category is called the Hom-category associated with monoidal category .
In the following, we recall from  some definitions about Hom-structures.
Definition 14. A unital monoidal Hom-associative algebra (which is called monoidal Hom-algebra in [16, Proposition 2.1]) is a vector space together with an element and linear maps such that for all .
Throughout, we use the conception of  for convenience. The definition of monoidal Hom-associative algebras is different from the unital Hom-associative algebras in [8, 19] in the following sense. The same twisted associativity condition (42) holds in both cases. However, the unitality condition on unital Hom-associative algebras is the usual untwisted one: , for any , and the twisting map does not need to be monoidal (i.e., (43) and (45) are not required).
In the language of Hopf algebras, is called the Hom-multiplication, is the twisting automorphism, and is the unit. Let and be two monoidal Hom-associative algebras. A monoidal Hom-algebra map is a linear map such that , , and .
Definition 15. A counital monoidal Hom-coassociative coalgebra (which is called monoidal Hom-coalgebra in [16, Prposition 2.4]) is a vector space together with linear maps , and such that for all .
Note that (46) is equivalent to , which is used often in the rest of the paper. Let and be two monoidal Hom-coassociative coalgebras. A monoidal Hom-coalgebra map is a linear map such that , , and .
The definition of monoidal Hom-coassociative coalgebra here is somewhat different from the counital Hom-coassociative coalgebra in [8, 19]. Their coassociativity condition is twisted by some endomorphism, not necessarily by the inverse of the automorphism . The counitality condition is the usual untwisted one. Counital Hom-coassociative coalgebras are not monoidal; that is, (47) and (49) are not required.
Definition 16. Let be a monoidal Hom-algebra. A right -Hom-module consists of in together with a morphism , such thatfor all and .
Similarly, we can define left -Hom-modules. Monoidal Hom-algebra can be considered as a Hom-module on itself by the Hom-multiplication. Let , be two left -Hom-modules. A morphism is called left -linear if , for any , , and . We denote the category of left -Hom-modules by .
is called an -Hom-bimodule if is both a left -Hom-module and right -Hom-module satisfying the following compatibility condition: for any .
Lemma 17. Let be a monoidal Hom-algebra, and let be a finite-dimensional -Hom-bimodule. Then there is an -Hom-bimodule structure on the dual of , where the actions are given by for any , , , and the set of integer numbers.
Proof. The rationality, unity, associativity, and the compatibility of left and right actions on are all derived from the same properties of -Hom-bimodule . This proof is left to the readers.
Due to the selection of in Lemma 17, there is a family of Hom-bimodule structures on of . In particular, the dual of can be considered as an -Hom-bimodule by selection or : for any , .
Definition 18. Let be a monoidal Hom-algebra and . is called a solution of the associative Hom-Yang-Baxter equation (AHYBE for short), if is -invariant, that is, , and satisfies the AHYBE where , , and .
Explicitly, the AHYBE in (55) can be rewritten as where is another copy of .
For a finite-dimensional monoidal Hom-algebra , any -invariant element can be considered as a morphism from to in by And the -invariance of implies ; that is, and . Further, if is symmetric (resp., skew-symmetric), then (resp., ).
Definition 19. Let be a monoidal Hom-algebra and . And let be an -Hom-bimodule. A morphism in is called an -operator of weight 0 associated with if it satisfies for any .
Particularly, is an -Hom-bimodule on itself. Then is called a Rota-Baxter operator of weight 0 if for any .
A Rota-Baxter operator is associated with Rota-Baxter algebra  which was studied further by Guo.
Proposition 20. Let be a finite-dimensional monoidal Hom-algebra and . If is skew-symmetric, then is a solution of AHYBE if and only if it is an -operator of weight associated with .
Proof. The -invariance of is equivalent to by the definitions.
For any , Similarly, we have So, . Thus and the multiplicativity of imply that is a solution of AHYBE if and only if it is an -operator of weight associated with .
6. Frobenius Monoidal Hom-Algebras and Hom--Operators
In this section, we consider the relationship between -operators (Rota-Baxter operators) and the AHYBE on Frobenius monoidal Hom-algebras.
Definition 21. A finite-dimensional monoidal Hom-algebra is called Frobenius if is an isomorphism as right -Hom-modules.
Lemma 22 (see [23, Proposition 5.2]). For a finite-dimensional monoidal Hom-algebra , the following assertions are equivalent:(1) is Frobenius.(2) as left -Hom-modules.(3)There exist elements , and such that , , and .(4)There exists a monoidal Hom-coalgebra , such that is an -Hom-bimodule morphism.(5)There exists a Hom-associative, nondegenerate bilinear form for . That is, there exists a bilinear map in , such that , and if or for any , then .
Definition 23. A Frobenius monoidal Hom-algebra is called symmetric if, for any , where is the nondegenerate Hom-associative bilinear form of the Frobenius monoidal Hom-algebra .
A linear map in is called self-adjoint (resp., skew-adjoint) with respect to a bilinear form if for any , (resp., ).
Then there is an induced injective linear map defined by the bilinear form for : for any . being a morphism in implies that is too; that is, . For a given Frobenius monoidal Hom-algebra , we always assume that is the induced morphism given by (63).
Lemma 24. Let be a finite-dimensional monoidal Hom-algebra and let be symmetric and -invariance. Then the following assertions are equivalent.(1) is invariant; that is, for any , (2) regarded as a linear map from to is balanced; that is, for any , (3) regarded as a linear map from to is an -Hom-bimodule homomorphism; that is, for any , ,
Proof. “(1)(2).” For any , , we have where the second step is followed by the actions and of on and the symmetry of , and the last step is true by the -invariance of . So is invariant if and only if it is balanced.
“.” For any , , we have where the third step is followed by the -invariance of and axiom (51) of action . So if and only if . Similarly, if and only if , for any , . Thus, is invariant if and only if regarded as a linear map from to is an -Hom-bimodule homomorphism.
Proposition 25. Let be a finite-dimensional monoidal Hom-algebra, let be symmetric and invariant. Suppose that regarded as a linear map from to in by is invertible. Then regarded as a bilinear form through equality (63) for is symmetric, nondegenerate, and Hom-associative. Thus is a symmetric Frobenius monoidal Hom-algebra.
Proof. Since is symmetric and regarded as a linear map from to in is invertible, is symmetric and nondegenerate. In addition, the invariance of implies that is balanced by Lemma 24. Then for any , , , and , we have where the third step is from the equality followed by the symmetry of , and the last step is true by the fact that is symmetric. Thus is Hom-associative, and is a symmetric Frobenius monoidal Hom-algebra.
Lemma 26. Let be a symmetric Frobenius monoidal Hom-algebra of finite dimension. Suppose that in is self-adjoint with respect to . Then regarded as an element of is symmetric, where is defined by equality (63). Moreover, is a balanced -Hom-bimodule homomorphism if and only if is a balanced -Hom-bimodule homomorphism.
Proof. For any , set , , and